A CLT for linear spectral statistics of large random information-plus-noise matrices
Résumé
Consider a matrix ${\rm Y}_n= \frac{\sigma}{\sqrt{n}} {\rm X}_n +{\rm A}_n, $ where $\sigma>0$ and ${\rm X}_n=(x_{ij}^n)$ is a $N\times n$ random
matrix with i.i.d. real or complex standardized entries and ${\rm A}_n$ is a $N\times n$ deterministic matrix with bounded spectral norm.
The fluctuations of the linear spectral statistics of the eigenvalues:
$$
\mathrm{Trace}\, f({\rm Y}_n {\rm Y}_n^*) = \sum_{i=1}^N f(\lambda_i),\quad (\lambda_i)\ \mathrm{eigenvalues\ of}\ {{\rm Y}}_n {{\rm Y}}_n^*,
$$
are shown to be gaussian, in the case where $f$ is a smooth function of class $C^3$ with bounded support, and in the regime where both dimensions of matrix ${{\rm Y}}_n$ go to infinity at the same pace.
The CLT is expressed in terms of vanishing Lévy-Prohorov distance between the linear statistics' distribution and a centered Gaussian probability distribution, the variance of which depends upon $N$ and $n$ and may not converge.
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